## On permutation

For any permutation define its displacement  as

What is greater: the sum of displacements of even permutations or the sum of displacements of odd permutations? The answer may depend on .

Solution

The sum of over the even permutations minus the one over the odd permutations is the determinant of the matrix with entries and this determinant is known to be

## Equal determinants

Let that are diagonizable in . If and   then prove that

Solution

The key point here is that the matrices are simultaneously diagonisable. Thus , there exists an invertible matrix and diagonisable matrices such that

Hence

where are the diagonial elements of and of . According to we have that

But , so and finally we conclude that:

Let be column vectors in and let be the corresponding real matrix. Then the following inequality holds:

(1)

where is the Euclidean norm on vectors in . In continuity , give the geometrical interpretation of the inequality above.

Solution

By the Gramm – Schmidt process we can establish the existence of an orthonormal basis such that

(2)

for each . Now, we may write for the corresponding real and orthogonal matrix. By orthogonality each vector in has an expansion as:

On the other hand implies that each vector has a shorter expansion of the form:

(3)

Alternatively let be the upper triangular matrix defined as:

Then is restated as and using again the fact that has orthonormal columns and the fact that is upper triangular we get:

Notes:

• The above argument also shows that there exists equality if and only if

for each . That is , if and only if, . This can only be achieved if the vectors are pairwise orthogonal.

• The geometrical interpretation of this inequality is the following: The volume of an dimensional parallelepiped produced by vectors can not exceed the product of their measures.

## No rational function

Prove that there exists no rational function such that

Solution

Suppose , on the contrary , that such function exists. Since the harmonic series diverges we conclude that the limit of our function in infinity is infinity. This, in return means that the degree of the nominator , call that is greater that the one of the denominator , call that . Extracting in the nominator we get that

The limit of at infinity is finite and call that . Hence:

and of course this contradicts the fact that

where is the Euler  – Mascheroni constant .

## Non existence of sequence of continuous functions

Prove that there does not exist a sequence of continuous functions such that converges pointwise, to the function , where is the characteristic polynomial of the rationals in .

Solution

The indicator of the rationals is no other function than

It is known that pointwise limits of continuous functions have a meagre set of points of continuity. However, this function is discontinuous everywhere and thus we cannot expect a sequence of continuous functions to converge pointwise to it.